Question: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $15.5$ years; the standard deviation is $2.2$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living between $19.9$ and $22.1$ years.
$15.5$ $13.3$ $17.7$ $11.1$ $19.9$ $8.9$ $22.1$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $15.5$ years. We know the standard deviation is $2.2$ years, so one standard deviation below the mean is $13.3$ years and one standard deviation above the mean is $17.7$ years. Two standard deviations below the mean is $11.1$ years and two standard deviations above the mean is $19.9$ years. Three standard deviations below the mean is $8.9$ years and three standard deviations above the mean is $22.1$ years. We are interested in the probability of a seal living between $19.9$ and $22.1$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the seals will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the seals will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of seals between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular seal living between $19.9$ and $22.1$ years is $\color{orange}{2.35\%}$.